They then select a set \(A_1\subset \bar{A}\)  with \(\#A_1\gtrapprox \#\bar{A}\) , so that \(A_1+A_1\)  has small \(\rho \) -covering number for all \(\delta \le \rho \le 1\) . The key estimates we need are the following analogues of Lemmas 3.6 and 3.7 from [1]}. Let \(a_1,a_2,b_2,b_2\in A_1\)  and let \(d_1=a_1-b_1,\)  \(d_2 = a_2-b_2\) . Then
\(\mathcal {E}_{\delta }(d_1A+ d_2A)\lessapprox K_1K_2K_3^{8}K_4^4\rho ^{4} \max (|d_1|,|d_2|)^{\alpha }\#(A),\) 
